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Splittings of finitely generated groups over two-ended subgroups

Splittings of finitely generated groups over two-ended subgroups
Splittings of finitely generated groups over two-ended subgroups
We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated groups which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.
0002-9947
1049-1078
Bowditch, Brian H.
559a0b03-4ffd-49b0-aafe-4764e7de5143
Bowditch, Brian H.
559a0b03-4ffd-49b0-aafe-4764e7de5143

Bowditch, Brian H. (1999) Splittings of finitely generated groups over two-ended subgroups. Transactions of the American Mathematical Society, 354 (3), 1049-1078.

Record type: Article

Abstract

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated groups which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.

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Published date: 1999

Identifiers

Local EPrints ID: 29774
URI: http://eprints.soton.ac.uk/id/eprint/29774
ISSN: 0002-9947
PURE UUID: ef6667b1-c38e-4dbb-b42c-0660a19405e0

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Date deposited: 22 Dec 2006
Last modified: 22 Jul 2020 16:51

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Author: Brian H. Bowditch

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